Figure 1.1 Shearing stresses in fluids

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-
Time Signal Processing
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Ronald W. Schafer

Figure 11.1

Linear system model for a signal
s
[
n
].


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-
Time Signal Processing
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Figure 11.2

Inverse filter formulation for all
-
pole signal modeling.


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-
Time Signal Processing
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Ronald W. Schafer

Figure 11.3

Linear prediction formulation for all
-
pole signal modeling.


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.4

Linear system model for a random signal
s
[
n
].


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-
Time Signal Processing
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Ronald W. Schafer

Figure 11.5

Examples of deterministic and random outputs of a 1
st
-
order all
-
pole system.


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-
Time Signal Processing
, Third Edition

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Ronald W. Schafer

Figure 11.6

Illustration (for
p
= 5) of computation of prediction error for the autocorrelation method. (Square dots
denote samples of
h
A
[
n
−
m
] and light round dots denote samples of
s
[
m
] for the upper plot and
e
[
n
] for the lower plot.)


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.7

Illustration of computation of the autocorrelation function for a finitelength sequence. (Square dots denote
samples of
s
[
n
+
m
], and light round dots denote samples of
s
[
n
].)


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.8

Illustration (for
p
= 5) of computation of prediction error for the covariance method. (In upper plot, square
dots denote samples of
h
A
[
n
−
m
], and light round dots denote samples of
s
[
m
].)


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.9

Illustration of computation of covariance function for a finite
-
length sequence. (Square dots denote
samples of
s
[
n
−
k
] and light round dots denote samples of
s
[
n
−
i
].)


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.10

Normalized mean
-
squared prediction error V
(p)

as a function of model order
p
in Example 11.2.


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.11

(a) Windowed voiced speech waveform. (b) Corresponding autocorrelation function (samples connected
by straight lines).


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.12

(a) Comparison of DTFT and all
-
pole model spectra for voiced speech segment in Figure 11.11(a). (b)
Normalized prediction error as a function of
p
.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.13

(a) Windowed unvoiced speech waveform. (b) Corresponding autocorrelation function (samples
connected by straight lines).


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.14

(a) Comparison of DTFT and all
-
pole model spectra for unvoiced speech segment in Figure 11.13(a). (b)
Normalized prediction error as a function of
p
.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.15

Zeros of prediction error filters (poles of model systems) used to obtain the spectrum estimates in

Figure 11.12.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.16

Spectrum estimation for a sinusoidal signal.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.17

Equations defining the Levinson
–
Durbin algorithm.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.18

Comparison of the Levinson
–
Durbin algorithm and the algorithm for converting from
k
-
parameters of a
lattice structure to the FIR impulse response coefficients in Eq. (11.85).


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.19

Signal flow graph of prediction error computation.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.20

Signal flow graph of lattice network implementation of
p
th
-
order prediction error computation.


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Discrete
-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure 11.21

All
-
pole lattice system.


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure P11.9


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-
Time Signal Processing
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Ronald W. Schafer

Figure P11.12
-
1


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-
Time Signal Processing
, Third Edition

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Ronald W. Schafer

Figure P11.12
-
2


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-
Time Signal Processing
, Third Edition

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Ronald W. Schafer

Figure P11.12
-
3


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Figure P11.17


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-
Time Signal Processing
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Ronald W. Schafer

Figure P11.18


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-
Time Signal Processing
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Ronald W. Schafer

Figure P11.19


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-
Time Signal Processing
, Third Edition

Alan V. Oppenheim •

Ronald W. Schafer

Table 11.1

PREDICTION COEFFICIENTS FOR A SET OF LINEAR PREDICTORS


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-
Time Signal Processing
, Third Edition

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Ronald W. Schafer

Figure P11.25
-
1


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-
Time Signal Processing
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Figure P11.25
-
2

Lattice structure for 2
nd
-
order system


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-
Time Signal Processing
, Third Edition

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Ronald W. Schafer

Figure P11.25
-
3

Lattice structure for 4
th
-
order system


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-
Time Signal Processing
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Ronald W. Schafer

Figure P11.27
-
1


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-
Time Signal Processing
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Figure P11.27
-
2